Maths Olympiad Prep

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Combinatorics Difficulty 1.4 Junior Find the answer Canada

In the diagram, six 4×44\times 4 squares have a percentage of their area shaded.

Figure 0

How many of
the 4×44\times 4 squares have exactly
25%25\% of their area shaded?

Pick one

Solution

The top-left square is divided into 44 regions having equal area. Since 11 of these 44 regions is shaded, then 25%25\% of the square is shaded.

The top-middle square is divided into 88 regions having equal area. Since 22 of these 88 regions are shaded, then 25%25\% of the square is shaded.

The top-right square is divided into 22 larger regions having equal areas, and
22 smaller regions having equal
areas. Since 11 of the larger
regions is shaded, then more than 25%25\% of the square is shaded.
Alternatively, the shaded region consists of three complete 1×11\times1 squares together with 66 half 1×11\times1 squares, and thus the shaded
area is 66. The entire 4×44\times4 square has area 1616, and 616\frac{6}{16} is not equivalent to 25%25\%.

The bottom-left square is divided into 1616 regions having equal area. Since 55 of these 1616 regions are shaded, then more than
25%25\% of the square is shaded.

The bottom-middle square is divided into 44 larger regions having equal areas, and
44 smaller regions having equal
areas. Since 22 of the 44 larger regions are shaded, and 22 of the 44 smaller regions are shaded, then 50%50\% of the square is shaded.

In the bottom-right square, consider constructing a vertical line
segment between the midpoint of the top side of the square and the
midpoint of the bottom side of the square. The square would then be
divided into 44 regions having equal
areas. Since 11 of these regions is
shaded, then 25%25\% of the square is
shaded.

Thus, the top-left, top-middle, and bottom-right squares have exactly
25%25\% of their area shaded and so
there are 33 such squares.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.